1. 7.2 The Law of Cosines and Area Formulas
SAS or SSS forms a unique triangle
Triangle Side Restriction
In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

2. 7.2 Derivation of the Law of Cosines
Let ABC be any oblique triangle drawn with its vertices
labeled as in the figure below.
The coordinates of point A become (c cos B, c sin B).
Figure 10 pg 10-28

3. 7.2 Derivation of the Law of Cosines
Point C has coordinates (a, 0) and AC has length b.
This result is one form of the law of cosines. Placing A
or C at the origin would have given the same result, but
with the variables rearranged.

4. 7.2 The Law of Cosines The Law of Cosines
In any triangle ABC, with sides a, b, and c,

5. 7.2 Using the Law of Cosines to Solve a Triangle (SAS)
Example Solve triangle ABC if
A = 42.3°, b = 12.9 meters, and
c = 15.4 meters.
Solution Start by finding a using the law of cosines.

6. 7.2 Using the Law of Cosines to Solve a Triangle (SAS)
B must be the smaller of the two remaining angles
since it is opposite the shorter of the two sides b and c.
Therefore, it cannot be obtuse.
Caution If we had chosen to find C rather than B, we would not have known whether C equals 81.7° or its supplement, 98.3°.

7. 7.2 Using the Law of Cosines to Solve a Triangle (SSS)
Example Solve triangle ABC if a = 9.47 feet,
b =15.9 feet, and c = 21.1 feet.
Solution We solve for C, the largest angle, first. If
cos C < 0, then C will be obtuse.

8. 7.2 Using the Law of Cosines to Solve a Triangle (SSS)
Verify with either the law of sines or the law of cosines
that B  45.1°. Then,

9. 7.2 Summary of Cases with Suggested Procedures
Oblique Triangle
Suggested Procedure for Solving
Case 1: SAA or ASA
Find the remaining angle using the angle sum formula (A + B + C = 180°).
Find the remaining sides using the law of sines.
Case 2: SSA
This is the ambiguous case; 0, 1, or 2 triangles.
Find an angle using the law of sines.
Find the remaining angle using the angle sum formula.
Find the remaining side using the law of sines.
If two triangles exist, repeat steps 2 and 3.

10. 7.2 Summary of Cases with Suggested Procedures
Oblique Triangle
Suggested Procedure for Solving
Case 3: SAS
Find the third side using the law of cosines.
Find the smaller of the two remaining angles using the law of sines.
Find the remaining angle using the angle sum formula.
Case 4: SSS
Find the largest angle using the law of cosines.
Find either remaining angle using the law of sines.

11. 7.2 Area Formulas
The law of cosines can be used to derive a formula for the area of a triangle given the lengths of three sides known as Heron’s Formula.
Heron’s Formula
If a triangle has sides of lengths a, b, and c, and if the semiperimeter is
Then the area of the triangle is

12. 7.2 Using Heron’s Formula to Find an Area
Example The distance “as the crow flies” from Los Angeles
to New York is 2451 miles, from New York to Montreal is 331
miles, and from Montreal to Los Angeles is 2427 miles. What is
the area of the triangular region having these three cities as
vertices? (Ignore the curvature of the earth.)
Solution

13. 7.2 Area of a Triangle Given SAS
The area of any triangle is given by A = ½bh, where b is its base and h is its height.
Area of a Triangle
In any triangle ABC, the area A is given by any of the following:

14. 7.2 Finding the Area of a Triangle (SAS)
Example Find the area of triangle ABC in the figure.
Solution We are given B = 55°, a = 34 feet, and
c = 42 feet.